Question

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

f(x) = ln {1 + e^x/ 1 - e^x}

Answer 1

The derivative of the function :
`f(x) = \ln (1 + e^x)/(1 - e^x)`

`f(x)'=(2e^x)/(1-e^(2x))`

Explanation:

`f(x) = \ln (1 + e^x)/(1 - e^x)`

Identity =
`(d(\ln x))/(dx)=(1)/(x)`

Quotient rule =
`(d((u)/(v)))/(dx)=(v(du)/(dx)-u(dv)/(dx))/(v^2)`

`f(x)'=(1)/((1 + e^x)/(1 - e^x))* ((1 - e^x)e^x-(1 + e^x)(-e^x))/((1 - e^x)^2)* 1`

`=(1)/((1 + e^x)/(1 - e^x))* (e^x(1-e^x+1+e^x))/((1-e^x)^2)`

`=(1)/((1 + e^x)/(1 - e^x))* (e^x(2))/((1 - e^x)^2)`

`=(1)/((1 + e^x)(1 - e^x))* 2e^x`

Identity =
`(a+b)(a-b)=a^2-b^2`

`=(2e^x)/(1-e^(2x))`